Found problems: 15925
2004 Cuba MO, 8
Determine all functions $f : R_+ \to R_+$ such that:
a) $f(xf(y))f(y) = f(x + y)$ for $x, y \ge 0$
b) $f(2) = 0$
c) $f(x) \ne 0$ for $0 \le x < 2$.
1992 IMO Longlists, 8
Given two positive real numbers $a$ and $b$, suppose that a mapping $f : \mathbb R^+ \to \mathbb R^+$ satisfies the functional equation
\[f(f(x)) + af(x) = b(a + b)x.\]
Prove that there exists a unique solution of this equation.
2011 Denmark MO - Mohr Contest, 4
A function $f$ is given by $f(x) = x^2 - 2x$ .
Prove that there exists a number a which satisfies $f(f(a)) = a$ without satisfying $f(a) = a$ .
2020 Grand Duchy of Lithuania, 1
Find all functions $f: R \to R$, such that equality $f (xf (y) - yf (x)) = f (xy) - xy$ holds for all $x, y \in R$.
2023 German National Olympiad, 4
Determine all triples $(a,b,c)$ of real numbers with
\[a+\frac{4}{b}=b+\frac{4}{c}=c+\frac{4}{a}.\]
2014 Contests, 1
Three positive real numbers $a,b,c$ are such that $a^2+5b^2+4c^2-4ab-4bc=0$. Can $a,b,c$ be the lengths of te sides of a triangle? Justify your answer.
VMEO III 2006, 12.3
Prove that for all $n\in\mathbb{Z}^+$, we have \[ \sum\limits_{p=1}^n\sum\limits_{q=1}^p\left\lfloor -\frac{1+\sqrt{8q+(2p-1)^2}}{2}\right\rfloor =-\frac{n(n+1)(n+2)}{3} \]
1988 Austrian-Polish Competition, 1
Let $P(x)$ be a polynomial with integer coefficients. Show that if $Q(x) = P(x) +12$ has at least six distinct integer roots, then $P(x)$ has no integer roots.
2013 IFYM, Sozopol, 6
The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that
$f(x+1)\leq f(2x+1)$ and $f(3x+1)\geq f(6x+1)$ for $\forall$ $x\in \mathbb{R}$.
If $f(3)=2$, prove that there exist at least 2013 distinct values of $x$, for which $f(x)=2$.
2011 AMC 12/AHSME, 19
At a competition with $N$ players, the number of players given elite status is equal to \[2^{1+\lfloor\log_2{(N-1)}\rfloor} - N. \] Suppose that $19$ players are given elite status. What is the sum of the two smallest possible values of $N$?
$ \textbf{(A)}\ 38\qquad
\textbf{(B)}\ 90 \qquad
\textbf{(C)}\ 154 \qquad
\textbf{(D)}\ 406 \qquad
\textbf{(E)}\ 1024$
1998 Chile National Olympiad, 7
When rolling two normal dice, the set of possible outcomes of the sum of the points is $2, 3, 3, 4,4, 4,..., 11, 11,12$. Notice that this sequence can be obtained from the identity $$(x + x^2 + x^3 + x^4 + x^5 + x^6) (x + x^2 + x^3 + x^4 + x^5 + x^6) = x^2 + 2x^3 + 3x^4 +... + 2x^{11} + x^{12}.$$ Design a crazy pair of dice, that is, two other cubes, not necessarily the same, with a natural number indicated on each face, such that the set of possible results of the sum of its points is equal to of two normal dice.
2013 Costa Rica - Final Round, 5
Determine the number of polynomials of degree $5$ with different coefficients in the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ such that they are divisible by $x^2-x + 1$. Justify your answer.
1977 Bundeswettbewerb Mathematik, 3
The number $50$ is written as a sum of several positive integers (not necessarily distinct) whose product is divisible by $100.$ What is the largest possible value of this product?
2015 Mexico National Olympiad, 3
Let $\mathbb{N} =\{1, 2, 3, ...\}$ be the set of positive integers. Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a function that gives a positive integer value, to every positive integer. Suppose that $f$ satisfies the following conditions:
$f(1)=1$
$f(a+b+ab)=a+b+f(ab)$
Find the value of $f(2015)$
Proposed by Jose Antonio Gomez Ortega
2004 Uzbekistan National Olympiad, 3
Given a sequence {$a_n$} such that $a_1=2$ and for all positive integer $n\geq 2$ $a_{n+1}=\frac{a_n^4+9}{16a_n}$. Prove that $\frac {4}{5}<a_n<\frac {5}{4}$
1995 All-Russian Olympiad, 2
Prove that every real function, defined on all of $\mathbb R$, can be represented as a sum of two functions whose graphs both have an axis of symmetry.
[i]D. Tereshin[/i]
1989 IMO Longlists, 17
Let $ a \in \mathbb{R}, 0 < a < 1,$ and $ f$ a continuous function on $ [0, 1]$ satisfying $ f(0) \equal{} 0, f(1) \equal{} 1,$ and
\[ f \left( \frac{x\plus{}y}{2} \right) \equal{} (1\minus{}a) f(x) \plus{} a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.\]
Determine $ f \left( \frac{1}{7} \right).$
2019 Israel National Olympiad, 1
In kindergarden, there are 32 children and three classes: Judo, Agriculture, and Math. Every child is in exactly one class and every class has at least one participant.
One day, the teacher gathered 6 children to clean up the classroom. The teacher counted and found that exactly 1/2 of the Judo students, 1/4 of the Agriculture students and 1/8 of the Math students are cleaning.
How many children are in each class?
1999 Harvard-MIT Mathematics Tournament, 10
$A, B, C, D,$ and $E$ are relatively prime integers (i.e., have no single common factor) such that the polynomials $5Ax^4 +4Bx^3 +3Cx^2 +2Dx+E$ and $10Ax^3 +6Bx^2 +3Cx+D$ together have $7$ distinct integer roots. What are all possible values of $A$?
[i]Your team has been given a sealed envelope that contains a hint for this problem. If you open the envelope, the value of this problem decreases by 20 points. To get full credit, give the sealed envelope to the judge before presenting your solution.[/i]
2021 Durer Math Competition (First Round), 4
Find all pairs of polynomials $(p, q)$ with integer coefficients that satisfy the equation $$p(x^2) + q(x^2) = p(x)q(x)$$ such that $p$ is of degree $n$ and has $n$ nonnegative real roots (with multiplicity).
1975 IMO Shortlist, 9
Let $f(x)$ be a continuous function defined on the closed interval $0 \leq x \leq 1$. Let $G(f)$ denote the graph of $f(x): G(f) = \{(x, y) \in \mathbb R^2 | 0 \leq$$ x \leq 1, y = f(x) \}$. Let $G_a(f)$ denote the graph of the translated function $f(x - a)$ (translated over a distance $a$), defined by $G_a(f) = \{(x, y) \in \mathbb R^2 | a \leq x \leq a + 1, y = f(x - a) \}$. Is it possible to find for every $a, \ 0 < a < 1$, a continuous function $f(x)$, defined on $0 \leq x \leq 1$, such that $f(0) = f(1) = 0$ and $G(f)$ and $G_a(f)$ are disjoint point sets ?
2019 Azerbaijan IMO TST, 1
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[ f(xy) = yf(x) + x + f(f(y) - f(x)) \]
for all $x,y \in \mathbb{R}$.
1974 IMO Longlists, 22
The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression \[ S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} \] takes?
2001 Manhattan Mathematical Olympiad, 4
How many digits has the number $2^{100}$?
2015 India Regional MathematicaI Olympiad, 3
Let $P(x)$ be a polynomial whose coefficients are positive integers. If $P(n)$ divides $P(P(n)-2015)$ for every natural number $n$, prove that $P(-2015)=0$.
[hide]One additional condition must be given that $P$ is non-constant, which even though is understood.[/hide]